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Design and Analysis of Optimized Selection Sort Algorithm Kirti Kaushik et al, International Journal of Computer Science and Mobile Computing, Vol.4 Issue.4, April- 2015, pg. 443-450 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 4, Issue. 4, April 2015, pg.443 – 450 RESEARCH ARTICLE Design and Analysis of Optimized Selection Sort Algorithm Kirti Kaushik* Roll No.15903, CS, Department of Computer science, Dronacharya College of Engineering, Gurgaon-123506, India Email: [email protected] Jyoti Yadav Roll No. 15040, CS, Department of Applied Computer science, Dronacharya College of Engineering, Gurgaon-123506, India Email: [email protected] Kriti Bhatia Roll No. 15048, CS, Department of Applied Computer science, Dronacharya College of Engineering, Gurgaon-123506, India Email: [email protected] ABSTRACT: One of the most frequent operations performed on database is searching. To perform this operation we have different kinds of searching algorithms, some of which are Binary Search, Index Sequential Access Method (ISAM), but these and all other searching algorithms work only on data, which are previously sorted. An efficient algorithm is required in order to make the searching algorithm fast and efficient. This research paper presents a new sorting algorithm named as “Optimized Selection Sort Algorithm, OSSA”.OSSA is designed to perform sorting quickly and more effectively as compared to the existing version of selection sort. The introduction of OSSA version of selection sort algorithm for sorting the data stored in database instead of existing selection sort algorithm will provide an opportunity to the users to save almost 50% of their operation time with almost 100% accuracy. INTRODUCTION One of the basic problems of computer science is ordering a list of items. There are a number of solutions to this problem, known as sorting algorithms. Some sorting algorithms are simple and spontaneous, such as the bubble sort. Others, such as the quick sort are enormously complex, but produce super-fast results. There are several elementary and advance sorting algorithms. All sorting algorithm are problem specific meaning they work well on some specific problem and do not work well for all the problems. All sorting algorithm are, therefore, appropriate for specific kinds of problems. Some sorting algorithm work on less number of elements, some are suitable for floating point numbers, some are good for specific range, some sorting algorithms are used for huge number of data, and some are used if the list has repeated values. We sort © 2015, IJCSMC All Rights Reserved 443 Kirti Kaushik et al, International Journal of Computer Science and Mobile Computing, Vol.4 Issue.4, April- 2015, pg. 443-450 data either in statistical order or lexicographical, sorting numerical value either in increasing order or decreasing order and alphabetical value like addressee key. The common sorting algorithms can be divided into two classes by the difficulty of their algorithms. There is a direct correlation between the complexity of an algorithm and its relative effectiveness. The complexity of algorithmic is generally written in a form known as Big – O (n) notation, where the O represents the complexity of the algorithm and a value n represents the size of the set the algorithm is run against. The two groups of sorting algorithms are O ( ), which includes the bubble, insertion, selection, and shell sorts; and O (n log n) which includes the heap, merge, and quick sort. Since the advancement in computing, much of the research is done to solve the sorting problem, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. It is always very difficult to say that one sorting algorithm is better than another. Performance of various sorting algorithms depend upon the data being sorted. Sorting is used in many important applications and there have been a plenty of performance analyses. However, earlier research is based on the algorithm’s theoretical complexity or their non-cached architecture. As almost all computers now a day’s contain cache, it is important to analyse them based on their cache performance. Quick sort was considered to be a good sorting algorithm in terms of average theoretical complexity and cache performance. Sorting is one of the most significant and well-studied subject area in computer science. Most of the first- class algorithms are known which offer various trade-offs in efficiency, simplicity, memory use, and other factors. However, these algorithms do not take into account features of modern computer architectures that significantly influence performance. A large number of sorting algorithms have been proposed and their asymptotic complexity, in terms of the number of comparisons or number of iterations, has been carefully analysed. In the recent past, there has been a growing interest on enhancements to sorting algorithms that do not have an effect on their asymptotic complexity but rather tend to improve performance by enhancing data locality. Sorting is an essential task that is performed by most computers. It is used commonly in a large variety of important applications. Database applications used by universities, banks, and other institutions all contain sorting code. Due to the importance of sorting in these applications, quite a large number of sorting algorithms have been developed over the decades with varying complexity. Some of the time consuming sorting methods, for example bubble sort,insertion sort, and selection sort have a hypothetical complexity of O( ). Although these algorithms are very slow for sorting larger amount of data, yet these algorithms are simple, so they are not useless. If an application only needs to sort smaller amount of data, then it is suitable to use one of the simple slow sorting algorithms as opposed to a faster, but more complicated sorting algorithm. ANALYSIS OF OLD SELECTION SORT ALGORITHM A. Selection Sort This is a very easy sorting algorithm to understand and is very useful when dealing with small amounts of data. However, as with Bubble sorting, a lot of data really slows it down. Selection sort does have one advantage over other sort techniques. Although it does many comparisons, it does the least amount of data moving. Thus, if your data has small keys but large data area, then selection sorting may be the quickest. B. Pseudo Code of Old Selection Sort Algorithm SelectionSort (X, n) =>X[0..n-1] 1. for i ← n – 1 to 0 1.1. IndexOfLarge ← 0 1.2. for j←1 to i 1.2.1 if (X[j]>X[IndexOfLarge)) 1.2.1.1 indexOfLarge ← j © 2015, IJCSMC All Rights Reserved 444 Kirti Kaushik et al, International Journal of Computer Science and Mobile Computing, Vol.4 Issue.4, April- 2015, pg. 443-450 1.3. Large ←X[IndexOfLarge] 1.4. X[IndexOfLarge] ← X[i] 1.5 X[i] ← Large C. Execution Flow Graph of Old Selection Sort D. Execution Time of Individual Statements In order to evaluate the execution time complexity of the given data of n elements. First we simplify the execution time of some inner loop statements in above algorithm. Note that ∑ ti,j = 1 when the if statement is true, 0 otherwise n-1 n-1 n-1 C3 Σ (i + 1) = C3 Σ i + C3 Σ 1 i=1 i=1 i=1 = C3 n(n – 1)/2 + C3 (n – 1) n-1 ∑ C4 Σ i = C4 n (n – 1)/2 i=1 n-1 ∑ C5 Σ i = C5 n (n – 1)/2, when ti, j =1 & 0 otherwise. © 2015, IJCSMC All Rights Reserved 445 Kirti Kaushik et al, International Journal of Computer Science and Mobile Computing, Vol.4 Issue.4, April- 2015, pg. 443-450 i=1 E. Best-Case Time Complexity of Old Selection Sort. Then for the best-case scenario we have that all ti,j = 0 so we get T(n) =C1 n + C2 n - C2 + C3 n (n – 1)/2 + C3 (n – 1)+C4 n (n – 1)/2+ C6n – C6 + C7n - C7 + C8 n - C8 T(n) =C1 n+ C2 n - C2 +C3 n2 /2 – C3 n/2 + C3 n – C3 + C4 n2/2– C4n/2 + C6 n – C6 + C7 n - C7 +C8 n - C8 T(n) = n2 (C3 / 2 + C4 /2) + n (C1 + C2 + C3 – C3 /2 - C4 /2+ C6 + C7 + C8) – (C2 + C3 +C7 +C8) Let a = (C3 / 2 + C4 /2) , b = (C1+C2+C3– C3/2 – C4/2+C6+ C7 + C8 )& c = – (C2 + C3 + C7 +C8) Then T(n ) becomes T(n) = a n2 + bn + c Thus here in best-case, the complexity of execution time of an algorithm shows the lower bound and is asymptotically denoted with Ω. Therefore by ignoring the constant a, b,c and the lower terms of n, and taking only the dominant term i.e. n2, then the asymptotic running time of selection sort will be Ω(n2) and will lie in of set of asymptotic function i.e Ө(n2). Hence we can say that the asymptotic running time of old SS will be: T(n) = Ө(n2) F. Worst - Case Time Complexity. Now for the worst-case scenario we have that all ti,j = 1 so we have T(n)=C1 n + C2 n - C2+ C3 n(n–1)/2 + C3 (n–1) +C4n (n–1)/2+C5n(n – 1)/2 + C6 n – C6+ C7 n - C7 + C8n - C8 T(n)=C1 n + C2 n - C2+ C3 n2 /2 – C3 n/2 + C3 n – C3 + C4 n2 /2– C4n/2 + C5 n2 /2– C5 n/2 +C6n – C6+C7n-C7 + C8 n - C8 T(n)= n2 (C3 / 2 + C4 /2+ C5 / 2) + n (C1 + C2 + C3 – C3 /2 – C4 /2 - C5 /2+C6 + C7 + C8 ) – (C2 + C3 + C6 + C7 +C8) Let a = (C3 / 2 + C4 / 2 + C6 / 2) b = (C1 + C2 + C3– C3 /2- C4 /2 - C5 /2+ C6 + C7 + C8) and c = – (C2 + C3 + C6 + C7 +C8) Then T(n ) becomes T(n) = a n2 + bn + c Thus here in worst-case, the complexity of execution time of an algorithm shows the upper bound and is asymptotically denoted with Big-O.
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